"The Following Are Equivalent
": an abbreviation
often used in writing Mathematics
(especially on a blackboard
Many times you have several "good" ways of defining the same thing. Which do you choose? It doesn't really matter, because the very next thing you'll find after the definition is:
- Original definition
- First restatement
- Second restatement...
So you could equally well have defined using (b) or (c) or ...
The claim of the theorem is that if any one of the conditions holds, then they all do.
To prove such a theorem, you don't have to show "(a) iff (b)" AND "(a) iff (c)"; it's enough to show that "if (a) then (b); if (b) then (c); if (c) then (a)". When you have many equivalent conditions, this saves a great deal of work.